Narrow Results

We investigate the impact of capital gains taxes on optimal investment decisions in a quite simple model. Namely, we consider a risk-neutral investor in the Black–Scholes model who owns one risky stock and determine the optimal stopping time at which he/she sells the stock to invest the proceeds in the bank account up to the maturity date. At the time the stock is sold, the investor has to realize book profits which triggers tax payments. For a linear tax, we derive a boundary that is continuous and increasing in time and decreasing in the volatility of the stock such that the investor sells the stock at the first time its price is smaller or equal to this boundary. For a variant of the problem with an exponentially distributed time horizon, we determine the boundary explicitly. In addition, we show how the structure of the stopping region changes if there are no tax credits for realized losses in the stock. Some numerical examples are given to exemplify the results.

Recently the new technique to solve optimal stopping problems for Hunt processes is developed (see [S. Christensen, P. Salminen and B. Q. Ta, Optimal stopping of strong Markov processes, Stochastic Process. Appl. 123(3) (2013) 1138–1159]). The crucial feature of the approach is to utilize the representation of the r-excessive functions as expected suprema. However, it seems to be difficult when applying directly the approach to some concrete cases, e.g. one-sided problem for reflecting Brownian motion and two-sided problem for Brownian motion. In this paper, we review and exploit this approach to find explicit solutions of two problems above.

Many companies issue some complex structured bonds. A reverse convertible bond is one of such structured bonds. In this paper we consider a valuation model of callable-puttable reverse convertible bonds which have the complex payoff in a setting of the optimal stopping problem between the issuer and the investor. Reverse convertible bonds issued by a company can be exchanged for the shares of another company. We analyze the pricing of reverse convertible bonds with call and put clauses and explore analytical properties of the value of the reverse convertible bond and optimal call and put boundaries by the issuer and the investor, respectively. Furthermore, we investigate how the call and put clauses affect the value and the optimal strategies for both of them.

The paper examines the optimal annuitization time and the optimal consumption/investment strategies for a retired individual subject to a constant force of mortality in an all-or-nothing framework. We allow for a different utility of consumption before and after annuitization. For a general family of preferences we characterize the value function and the optimal controls of the resulting combined stochastic control and optimal stopping problem. Assuming power utility functions we obtain explicit solutions. We show that if the individual evaluates the consumption flow and the annuity payments stream in the same way, then, depending on the parameters of the economy, the annuity is purchased at retirement or never. In the case when the individual is more risk averse in the annuity assessment, it is optimal to defer the annuitization until her wealth reaches a threshold, and such threshold depends on the parameters of the economy.

The aim of this work is to use a duality approach to study the pricing of derivatives depending on two stocks driven by a bidimensional Lévy process. The main idea is to apply Girsanov's Theorem for Lévy processes, in order to reduce the posed problem to a problem with one Lévy driven stock in an auxiliary market, baptized as "dual market". In this way, we extend the results obtained by Gerber and Shiu [5] for two-dimensional Brownian motion.

A canonical problem in real option pricing, as described in the classic text of Dixit and Pindyck [2], is to determine the optimal time to invest at a fixed cost, to receive in return a stochastic cashflow. In this paper we are interested in this problem in an incomplete market where the cashflow is not spanned by the traded assets. We follow the formulation in Miao and Wang [21]; our contribution is to show that significant progress can be made in solving the Hamilton-Jacobi-Bellman equation and that the optimal exercise threshold can be characterized quite precisely.

In this paper, we investigate the generalization of the Call-Put duality equality obtained in Alfonsi and Jourdain (preprint, 2006, available at ) for perpetual American options when the Call-Put payoff (y - x)⁺ is replaced by ϕ(x,y). It turns out that the duality still holds under monotonicity and concavity assumptions on ϕ. The specific analytical form of the Call-Put payoff only makes calculations easier but is not crucial unlike in the derivation of the Call-Put duality equality for European options. Last, we give some examples for which the optimal strategy is known explicitly.

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information. The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. We establish conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward, and derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not. Hence we show that early exercise may arise due to the agent having inside information on the future stock price.

The optimal capital structure model with endogenous bankruptcy was first studied by Leland (1994) and Leland & Toft (1996), and was later extended to the spectrally negative Lévy model by Hilberink Rogers (2002) and Kyprianou Surya (2007). This paper incorporates scale effects by allowing the values of bankruptcy costs and tax benefits to be dependent on the firm's asset value. By using the fluctuation identities for the spectrally negative Lévy process, we obtain a candidate bankruptcy level as well as a sufficient condition for optimality. The optimality holds in particular when, monotonically in the asset value, the value of tax benefits is increasing, the loss amount at bankruptcy is increasing, and its proportion relative to the asset value is decreasing. The solution admits a semi-explicit form in terms of the scale function. A series of numerical studies are given to analyze the impacts of scale effects on the bankruptcy strategy and the optimal capital structure.

In this paper, we consider the optimal execution problem associated to accelerated share repurchase (ASR) contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank.

This paper deals with optimal prediction in a regime-switching model driven by a continuous-time Markov chain. We extend existing results for geometric Brownian motion by deriving optimal stopping strategies that depend on the current regime state and prove a number of continuity properties relating to optimal value and boundary functions. Our approach replaces the use of closed form expressions, which are not available in our setting, with PDE arguments that also simplify the approach of [du Toit & Peskir (2009) Selling a stock at the ultimate maximum, Annals of Applied Probability19 (3), 983–1014.] in the classical Brownian case.

Following the economic rationale of Peskir & Samee [The British put option, Applied Mathematical Finance18 (6), 537–563 (2011); The British call option, Quantitative Finance13 (1), 95–109 (2013)], we present a new class of asset-or-nothing put option where the holder enjoys the early exercise feature of American asset-or-nothing put option whereupon his payoff is the ‘best prediction’ of the European asset-or-nothing put option payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, the option holder finds that if the true drift of the stock price is unfavorable, then he can substitute it with the contract drift and minimize his losses. The key to the British asset-or-nothing put option is the protection feature as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favorable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. We also analyze the financial meaning of the British asset-or-nothing put option using the results above and show that with the contract drift properly selected, the British asset-or-nothing put option becomes a very attractive alternative to the classic European/American asset-or-nothing put option.

We study the optimal timing strategies for trading a mean-reverting price process with a finite deadline to enter and a separate finite deadline to exit the market. The price process is modeled by a diffusion with an affine drift that encapsulates a number of well-known models, including the Ornstein–Uhlenbeck (OU) model, Cox–Ingersoll–Ross (CIR) model, Jacobi model, and inhomogeneous geometric Brownian motion (IGBM) model. We analyze three types of trading strategies: (i) the long–short (long to open, short to close) strategy; (ii) the short–long (short to open, long to close) strategy, and (iii) the chooser strategy whereby the trader has the added flexibility to enter the market by taking either a long or short position, and subsequently close the position. For each strategy, we solve an optimal double stopping problem with sequential deadlines, and determine the optimal timing of trades. Our solution methodology utilizes the local time-space calculus of [Peskir (2005) A change-of-variable formula with local time on curves, Journal of Theoretical Probability18, 499–535] to derive nonlinear integral equations of Volterra-type that uniquely characterize the trading boundaries. The numerical implementation of the integral equations provides examples of the optimal trading boundaries.

We introduce a new approach for systematically obtaining smooth deterministic upper bounds for the price function of American style options. These bounding functions are characterized by sufficient conditions, under which the bounds may be infimized. In a single implementation, the proposed approach obtains explicit bounds in the form of piecewise polynomial functions, which bound the price function from above over the whole problem domain both in time and state. As a consequence, these global bounds store a continuum of information in the form of a finite number of polynomial coefficients. The proposed approach achieves these bounds, free from statistical error, without recourse to sample path simulation, without truncating the naturally unbounded domain that arises in this problem, and without discretizing the time and state variables. Throughout the paper, we demonstrate the effectiveness of the proposed method in obtaining tight upper bounds for American style option prices in a variety of market models and with various payoff structures, such as the multivariate Black Scholes and Heston stochastic volatility models and the American put and butterfly payoff structures. We also discuss extensions of the proposed methodology to perpetual American style options and frameworks in which the underlying asset contains jumps.

The American chooser option is a relatively new compound option that has the characteristic of offering exceptional risk reduction for highly volatile assets. This has become particularly significant since the start of the global financial crisis. In this paper, we derive mathematical properties of American chooser options. We show that the two optimal stopping boundaries for American chooser options with finite horizon can be characterized as the unique solution pair to a system formed by two nonlinear integral equations, arising from the early exercise premium (EEP) representation. The proof of EEP representation is based on the method of change-of-variable formula with local time on curves. The key mathematical properties of American chooser options are proved, specifically smooth-fit, continuity of value function and continuity of free-boundary among others. We compare the performance of the American chooser option against the American strangle option. We also conduct numerical experiments to illustrate our results.

We study the optimal exercise of American options under incomplete information about the drift of the underlying process, and we show that quite unexpected phenomena may occur. In fact, certain parameter values give rise to stopping regions very different from the standard case of complete information. For example, we show that for the American put (call) option it is sometimes optimal to exercise the option when the underlying process reaches an upper (lower) boundary.

The Longstaff–Schwartz (LS) algorithm is a popular least square Monte Carlo method for American option pricing. We prove that the mean squared sample error of the LS algorithm with quasi-regression is equal to $c_1/N$ asymptotically,^a where $c_1>0$ is a constant, $N$ is the number of simulated paths. We suggest that the quasi-regression based LS algorithm should be preferred whenever applicable. Juneja & Kalra (2009) and Bolia & Juneja (2005) added control variates to the LS algorithm. We prove that the mean squared sample error of their algorithm with quasi-regression is equal to $c_2/N$ asymptotically, where $c_2>0$ is a constant and show that $c_2<c_1$ under mild conditions. We revisit the method of proof contained in Clément et al. [E. Clément, D. Lamberton & P. Protter (2002) An analysis of a least squares regression method for American option pricing, Finance and Stochastics, 6 449–471], but had to complete it, because of a small gap in their proof, which we also document in this paper.

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary.

To investigate the effect of short-selling constraints on investor behavior, we formulate an optimal stopping model in which the decision to cover a short position is affected by two short sale-specific frictions — margin risk and recall risk. Margin risk is introduced by assuming that a short seller is forced to close out their position involuntarily if they cannot fund margin calls (since short sales are collateralized transactions). Recall risk is introduced by permitting the lender to recall borrowed stock at any time, once again triggering an involuntary close-out. Examining the effect of these frictions on the optimal close-out strategy and associated value function, we finding that the optimal behavior can be qualitatively different in their presence. Moreover, these frictions lead to a substantial loss in value, relative to the first-best situation without them (a reduction of approximately 17% for our conservative base-case parameters). This significant effect has important implications for many familiar no-arbitrage identities, which are predicated on the assumption of unfettered short selling.

We consider a financial market in which the risk-free rate of interest is modeled as a Markov diffusion. We suppose that home prices are set by a representative homebuyer, who can afford to pay only a fixed cash flow per unit time for housing. The cash flow is a fraction of the representative homebuyer’s salary, which grows at a rate that is proportional to the risk-free rate of interest. As a result, in the long run, higher interest rates lead to faster growth of home prices. The representative homebuyer finances the purchase of a home by taking out a mortgage. The mortgage rate paid by the homebuyer is fixed at the time of purchase and equal to the risk-free rate of interest plus a positive constant. As the homebuyer can only afford to pay a fixed cash flow per unit time, a higher mortgage rate limits the size of the loan the homebuyer can take out. As a result, the short-term effect of higher interest rates is to lower the value of homes. In this setting, we consider an investor who wishes to buy and then sell a home in order to maximize his discounted expected profit. This leads to a nested optimal stopping problem. We use a nonnegative concave majorant approach to derive the investor’s optimal buying and selling strategies. Additionally, we provide a detailed analytic and numerical study of the case in which the risk-free rate of interest is modeled by a Cox–Ingersoll–Ross (CIR) process. We also examine, in the case of CIR interest rates, the expected time that the investor waits before buying and then selling a home when following the optimal strategies.